Master Thesis
“Along strike variation of active fault arrays and their effect on landscape morphology in the northwestern Himalaya”
Author: Markus Nennewitz 1. Examiner: Dr. Rasmus C. Thiede Mat-Nr.: 754570 2. Examiner: Prof. Dr. Bodo Bookhagen
Declaration of plagiarism
I hereby declare that this thesis is the result of my own independent work, and that in all cases material from the work of others (in books, articles, essays, dissertations, and on the internet) is acknowledged, and quotations and paraphrases are clearly indicated. No other material than that listed has been used.
1/62 Acknowledgements First of all I would like to thank my supervisors Dr. Rasmus Thiede and Prof. Dr. Bodo Bookhagen, who broadend my mind for active tectonics and geomorphology during many meetings and even more e-mail conversations. They supported me in every minute of my work and were always open for questions from my side. Furthermore, I thank them and the DAAD for giving me the opportunity to go on a field trip in NW India. According to that, I would like to mention that I really appreciated the stay at the IIT Gandhinagar for which I also thank Prof. Dr. Vikrant Jain and his Phd-students. Furthermore, I have to thank Saptarshi Dey and Patricia Eugster for providing data files necessary for my analysis. And last but definitely not least I thank my girlfriend and my parents for their tremendous amount of support and for being patient with me during my ongoing work.
2/62 Abstract The location and magnitude of the active deformation of the Himalaya has been in the interest for many decades. Still the understanding of the neotectonics and its effect on the regional topography is improvable. This study investigates the along strike changes of fault activity and segmentation in the northwestern Himalaya. Therefore we have performed a river network analysis and obtained the channel steepness indexes for tributaries with a drainage area falling in a range of 1km² to 100km². The indexes were averaged over catchments with a Strahler-order of 3. We used orogen-perpendicular and along strike profiles to determine areas of equal subsurface geometries and active fault segments within. The observed pattern of along strike variation in fault activity leads to the conclusion that three segments (A1-A3) operate independently from each other. A1 covers the Dehra Dun, the Nahan Salient and the Garwhal region. A2 is located in the Kangra Dun and the Chamba Himalaya. A3 contains the Kashmir Himalaya and the respective foreland fold- and thrust-belt. Despite the differences in the structural architecture of the orogenic front, we found good reason for an out-of-sequence activity of segments of the PT2 promoting structure, as well as of fault segments of the MBT in all three areas since the Pleistocene.
Zusammenfassung Das Verständniss von neotektonischen Prozessen im Nordwest Himalaya wurde bereits in vielen Studien untersucht. Dennoch haben wir Grund zur Annahme, dass dieses Verständniss ausbaufähig ist. Speziell die Aktivität von einzelnen Störungssegmenten entlang der MBT und der geologischen Struktur unterhalb der PT2 wurde noch nicht weitreichend analysiert. Wir haben die -Werte einzelner Segmente von Zuflüssen mit einer Einzugsgebiet von 1km² bis zu 100km² im Untersuchungsgebiet ermittelt. Wir haben die Werte für einzelne Wassereinzugsgebiete gemittelt und mithilfe von orthogonalen Profilen und Profilen, die parallel zum Streichen von Störungen verlaufen, analysiert und in den Zusammenhang gebracht. Durch diese Methode können wir Aussagen über die regionale Varianz in den Erosiosraten treffen, welche uns unter bestimmten Vorrausetzungen wiederum auch auf Unterschiede in der tektonischen Aktivität von Störungen schließen lässt. Im Zuge der Untersuchung stellte sich heraus, das es drei Gebiete im nordwest Himalaya gibt, die nur durch einen unterschiedlichen strukturgeologischen Aufbau zu erklären sind (A1-A3). Trotz dieses Unterschiedes fanden wir heraus, dass es im gesamten Untersuchungsgebiet entlang der Front des Hohen Himalaya und entlang der MBT eine out-of-sequence Aktivität von einzelnen Störungssegmenten gibt.
3/62 Table of Contents Abstract ...... 3 Introduction ...... 6 Geologic setting ...... 8 Tectonic evolution and recent setting ...... 8 Description of physiographic units ...... 11 Theories for the accommodation of the crustal shortening ...... 12 Direction and velocity of plate motion ...... 13 Seismic activity and geodetic velocities in NW India...... 14 Methods ...... 16 River steepness and concavity index ...... 16 Knickpoints in longitudinal river profiles ...... 18 Processing the channel steepness indexes ...... 19 Post-processing steepness indexes ...... 21 Correction for glacial erosion ...... 21 Catchment-wide steepness index ...... 21 Hot Spot Analysis ...... 22 Topometric analysis ...... 22 Orthogonal swath profiles ...... 22 Parallel swath profiles ...... 23 Results ...... 25 Channel steepness in the NW Himalaya ...... 25 Description of the perpendicular swath profiles ...... 26 Summary of profile 1-10 ...... 32 The cumulative height ...... 33 Description of fault-parallel swath profiles ...... 34 Discussion ...... 37 Significance in data distribution ...... 37 Applicability of the method ...... 38 Active out-of-sequence thrusting vs. elastic behavior of the upper plate ...... 40 Area A1 - A Central Himalayan analogue ...... 41 Area A2 – The Chamba Himalaya ...... 43 Area A3 – The Kashmir Himalaya ...... 44 Along strike variations ...... 46 Recent denudation at the PT2 ...... 46
4/62 Tectonic activity of MBT-Segments ...... 47 Total strain distribution ...... 49 Relationship to published rates ...... 50 Relation to exhumation and denudation rates ...... 52 Conclusion ...... 55 Reference...... 56
5/62 Introduction The continental collision of India and Eurasia has formed the highest recent orogen on earth, the Himalaya. The ongoing north-north-east oriented motion of the Indian plate still leads to active tectonics. Major thrust systems bounding the physiographic units of the Himalaya and the Tibetan plateau accommodate the resulting convergence (Wang et al. 2001). As a consequence, the landscape is forced to adjust to the tectonic conditions. Thus, the interplay between rock uplift and erosion, which are the key processes, forms the topography of the orogen. Researchers found ways to use this dependency in order to recalculate the erosion (or rock uplift) from the topography. A powerful tool is the analysis of the morphology of river channels which has proved to be a robust proxy for the occurring erosion. The methods implemented in the early 20th century have been developed and improved to better constrain the underlying processes e.g. (Hack 1957, 1973; Whipple und Tucker 1999a). Today many researchers perform river network analysis to gain the steepness and the concavity index of channel segments for assumptions about the recent tectonic conditions e.g.(Whittaker 2012). For the Himalaya, a tremendous amount of studies has been published in order to explain the recent tectonic mechanisms responsible for the uplift of the orogen and some of them by using the mentioned indexes. e.g. (Ader et al. 2012; Bollinger et al. 2006; Lavé & Avouac 2000, 2001; Wobus et al. 2005; Wobus et al. 2006b; Whipple et al. 2016; Elliott, J. R. et al. 2016). The Central Himalaya is attractive in many ways for such researches. One benefit derives from the direction of the plate convergence which is perpendicular to the mountain front. Therefore the expected pattern of the deformation is simpler than in oblique, compressional zones additionally experiencing shear motion. Furthermore, the region in Central Nepal is favored for this kind of analysis because the setting is representative for a large area of the Himalaya. The classic geometry of the orogenic wedge contains four pronounced physiographic units separated by major fault systems (Fig. 1 & 24) Nevertheless, seismic sections are lacking for large parts of the Himalaya. Thus, geometries of faults and other structural features are still unknown. There is good reason that the subsurface is more differentiated and segmented than expected as shown by Harvey et al. (2015) suggesting changes in the geometry of the subsurface of the Central Himalaya via analysis of river networks.
According to the northwestern part of the Himalaya, previous studies already described topographies of the mountain front deviating from the classic build up. e.g. (Burbank 1983; Thakur 1998; Powers et al. 1998).
We have expanded the approach of the river network analysis to the northwestern Himalaya. We calculated the river steepness indexes for tributaries with a drainage area falling in a range
6/62 between 5 km² and 200 km². We use this proxy for erosion in order to look for along strike changes in the thrust geometry and to assess the activity of fault segments as well as their contribution to the crustal shortening. The gained 2D-results display a coherent overview of the distribution of the erosion rate in NW India supporting an out-of-sequence activity of multiple fault segments. The expected tectonic activity of the MBT as well as of the PT2 promoting structure is variable along strike. These results improve the understanding of the ongoing orogenese and the accommodated deformation since the Pleistocene. Moreover it improves the understanding of the relationship between segments of different subsurface geometries.
7/62 Geologic setting
Tectonic evolution and recent setting The Himalaya is the highest orogen on earth. As such, a lot of research has been done in order to understand the process of this tremendous rise as a consequence of the continental collision of the Indian and the Eurasian tectonic plate. The tectonic evolution of the Himalaya is roughly identical along the whole mountain range. After the closure of the Neotethys ocean the collision of the continental plates India and Eurasia started 55 Myr ago in the NW Himalaya (Thakur 1993) p.319. As a consequence, the northern margin of the Indian plate was deformed and its rocks underwent regional metamorphism resulting in the units of the Greater Himalaya. As a result of the continued convergence, the Main Central Thrust (MCT) has uplifted those metamorphic units above the formations of the Lower (or Lesser) Himalaya (Thakur 1993) p.6. The southern limit of the Lower Himalaya is set by the MBT (Main Boundary Thrust), which developed at least 10 Ma ago (Meigs et al. 1995). Lying in between the MBT and the MCT, the Lower Himalaya had been folded and underwent a low grade of metamorphism. This very continuous thrust fault has overridden the units of the Subhimalaya which mainly comprises of molasse material (Thakur 1993) p.5. Drillings have shown that these lithologic units reach far south into the Ganga basin (Powers et al. 1998; Mugnier und Huyghe 2006). This supports the theory of a continuous foreland basin. Due to the propagation of the orogenic wedge in the late Quaternary, the Main Frontal Thrust (MFT) has developed as the youngest in-sequence thrust of the Himalaya (Thakur et al. 2007; Thakur 1993) p.5. The Subhimalaya zone can be best described as a foreland fold-and thrust-belt. The triplet of MCT, MBT and MFT, although deviating in distance to each other, can be found all along strike of the Himalayan orogen (Fig. 1).
8/62 In contrast to the Central Himalaya, previous studies observed that in the northwestern Himalaya the convergence direction is not perpendicular to the mountain front e.g. (Tapponnier und Molnar 1979; Kundu et al. 2014; Silver, Calvin R. P. et al. 2015). Thus, the oblique convergence results in strain partitioning where the arc-normal deformation is accommodated via thrusting along the main faults and a strike slip component which is accommodated in the orogen interior. One of the main agents accommodating the strike slip component and other exceptional feature in the NW Himalaya are described in more detail.
Figure 1 Geologic map of the Western Himalaya.(modified from Hodges, 2000) The red rectangle bounds the study area of this thesis.
Karakorum-Fault For the scope of this thesis, it is necessary to mention some regional fault systems which are only present in the north western part of the mountain front but may have an enormous effect on the tectonic evolution. The NW striking Karakorum fault is about 1000km in length and it is part of the contractional system in the western portion of the Himalayan-Tibetan orogenic belt. To the east, the oblique, dextral fault joins into the frontal thrusts of the
9/62 Himalaya. Its northern extension reaches into the Pamir orogen where it splays into dextral strike-slip and thrust faults (Strecker et al. 1995). Traced horizons like the Aghil-fm. cut by the fault revealed that the total offset of the Karakorum fault is between 149 km and 167 km (Robinson 2009a). According to performed GPS measurements, the fault rather pushes the southern Himalaya to the west than the Tibetan Plateau to the east (Banerjee 2002). Today’s velocity based on GPS-data is still debated and ranges from 1-4mm∙yr-1 (Banerjee 2002) to 5±2 mm∙yr-1(Kundu et al. 2014). However, the slip rate does not seem to be constant through the ages (Kundu et al. 2014; Banerjee 2002). Geological offsets expect rates of ca. 11mm∙yr-1 since the Miocene (Robinson 2009a). Geomorphic studies using cosmogenic nuclide dating of offset moraine surfaces have presented velocities of ~10 mm∙yr-1 (Chevalier et al. 2005). Due to the nature of this method, the rate is representative for the motion on a millennial timescale.
Duns and Salients In general, the collision of India and Asia resulted in parallel longitudinal ranges but looking closer the resulting thrust geometries in the northwestern Himalaya are slightly different. The MBT describes a rather sinuous trace in the NW part of the orogenic belt (Fig.1). In contrast, the NW striking MFT is rather straight. As a consequence of this difference the fold- and thrust belt changes its width and develops ‘Duns’ and ‘Salients’. Outstanding is the Kangra Dun (or Punjab re-entrant) having a maximal width of ca. 140km between the MBT and the MFT. Thus, the Kangra Dun is the largest reentrant to be found in NW India. It is bounded by the Nahan salient to the south and the outlet of the Ravi River to the north. The occurrence of duns and salients is still not completely understood but previous studies suggest that the angle of the MHT is higher beneath a salient (Singh et al. 2012).
Kashmir Basin The Kashmir Basin is located northwest of the Kangra reentrant and is a rather exceptional feature in the structure of the Himalaya (Fig. 1). As already mentioned the MBT was formed at least 10 Myr ago (Meigs et al., 1995). This is also true for northwestern part of the Himalaya but the thrust had been displaced towards the southwest at least 4 Myr ago and uplifted the southwestern margin of the basin, the Pir Panjal Range (Burbank & Johnson, 1982), (Burbank 1983). The active mountain front propagated further when the Medlicott-Wadia-Thrust and the Suruin-Mastgarth Anticline developed, most likely since 2 Ma (Burbank et al. 1986). The northeastern margin of the Kashmir Basin is set by the Greater Himalayan Range (Burbank & Johnson, 1982).
10/62 Description of physiographic units
Tethys Himalaya The Tethys Himalaya Zone is a metasedimentary layer on top of the Higher Himalayan metamorphic rocks with a maximal thickness of 10km. The marine sediments have been deposited between the late Precambrium to the Lower Eocene on a shelf or marine slope, the former northern passive margin of India (Thakur, 1993) p. 149. This enormous time span gave enough opportunity to deposit different kinds of siliciclastic and calcareous sediments. The zone is present from the Zanskar Mountains in the west along the whole southern margin of the Tibetan plateau (Thakur 1993) p.6. It is also known as the Tibetan Zone or Tibetan Himalayan Zone.
Higher Himalaya The crystalline band of the Higher Himalaya was uplifted via the MCT. Due to barrowvian metamorphism during early stages of Himalayan crustal thickening, rocks varying from green schist to highest amphibolite facies and migmatites are the most common (Thakur 1993) p.107. In addition, granites, granitoids and orthogneisses are also common because of several intrusive events in the Cambrian and in the Tertiary (Thakur 1993) p.107.
Lower Himalaya Proterozoic to Eocene rocks, covering the basement of the Indian craton, which have been detached by the underthrusting of the Indian plate and incooperated into the Himalayan orogenic wedge build the units of the Lower (or Lesser) Himalaya. They underwent a lower grade of metamorphism. It is mainly made of low grade metasediments. Sheared sediments like phyllites and schists but also calcareous rocks are common. Nevertheless, the erosional resistivity of the material is lower compared to the crystalline rocks of the Higher Himalaya.
Subhimalaya Completely unmetamorphosed, sedimentary rocks describe this sequence. It is limited by the MBT in the north and by the MFT to the south. The including formations have been continuously deposited in front of the whole mountain range. The first units of the Subathu group, marking a marine transgression and thus the closure of the Neotethys, were formed in the Upper Paleocene (Thakur 1993) p.20. Due to the shallow marine environment mainly fossil-bearing shales and limestones had been deposited (Thakur 1993) p.20. Since the late Eocene, the Dharamsala group follows in the area of Himachal Pradesh. This group is divided into two subgroups, the Lower and the Upper Dharamsala which have been deposited at the beginning of the Miocene. The Lower Dharamshala is characterized by “purple clays, siltstones and greenish grey and red clays”(Thakur 1993)p.20. Due to the rising supply of coarser
11/62 terrestrial material into the depositional environment during the continued orogenese, the Upper Dharamsala contains more “sandstone with minor amounts of greenish grey and red clays”(Thakur 1993) p.20. The upward coarsening trend continues in cycles through the Siwalik groups. The Lower Siwalik contains alternations of sand- and claystone. Some horizons contain limestone, quarzite and sandstone clasts (Thakur 1993). The Middle Siwalik mainly consists of sandy, arkosic litharenites alternating with minor claystone layers. Alternation with pebbly conglomerates can be observed in the upper part. The Upper Siwalik units have been deposited from the Pliocene to the Pleistocene. They are made of polymictic, boulder conglomerates with the occasionally occurrence of sand lenses. According to my observations in the field, the cementation of single beds alternates. In general, well cemented conglomerates can rather be found in the lower part of the group. The youngest formation is the Neogal. The sediments of this unit fill the piggy back basins of the internally deformed Siwalk units. The majority of the sediment supply of the Kangra basin derives from the Dhauladhar range which is mainly made of granitoid rocks. Thus, the alluvial fan deposits in front of the range and other filling material are made of monomictic conglomerates with a minor amount of mobilized clasts from the Siwalik group. These beds are usually less cemented than the Upper Siwalik beds.
Theories for the accommodation of the crustal shortening In the most classical structure of the Himalaya which is representative for large areas of the orogen, the four physiographic units, described above, are bounded by major fault systems (Fig. 1) (Gansser A. 1964), (Hodges 2000) and references therein). Our understanding and structural constrain of the geometry of the Himalayan thrust system throughout the entire orogen is still loose. The knowledge about the exact depth or location of major fault geometries and the magnitude of slip rates at major fault systems are still strongly debated. One of the most obvious topographic features has been the pronounced change in the topography, and steepened longitudinal river profiles along the transition between Lesser and Higher Himalaya (e.g Seeber & Gornitz 1983; Wobus et al. 2005). Therefore, the Central Himalaya has been in the focus of many studies explaining the physiographic transition between the Lower and the Higher Himalaya and how the convergence between the Indian plate and Eurasian plate is accommodated. Their structural architecture mostly agrees in terms of a ramp structure along the MHT (Fig. 2). However, three theories have developed explaining the deformation along the MHT. The first scenario is characterized by in-sequence thrusting (Fig. 2A). Here, the total amount of shortening is accommodated at the MFT (Lavé & Avouac 2000). This had been observed south of the Kathmandu Basin. The study determined the incision rate of the Bagmati and the Bakeya River to infer the necessary rock uplift rate of
12/62 Figure 2. The simplified models illustrate three different concepts of how the tectonic convergence is accommodated at the Himalayan front. (Figure modified from Wobus et al.,2006) Model A) Rock uplift around the PT2 is the result of material transport over a mid-crustal ramp. (Lavé & Avouac, 2000) Model B) Accretion of the footwall into the hanging wall forms a passive duplex structure accommodating the shortening (Bollinger et al. 2004; Wobus et al. 2006, Elliot et al. 2016). Model C) Active out- of-sequence thrusting results in uplift of the Greater Himalaya (Wobus et al., 2005; Whipple et al. 2016) the MFT. They claim that the MHT entirely ruptures at once resulting in very large earthquakes. The change in topography and relief at the PT2 is solely caused by the slip of the Eurasian plate over a mid-crustal ramp (Lavé & Avouac 2000; Herman et al. 2010). The second scenario explains the pronounced deformation around the PT2 by presence of a duplex structure developing by accretion of the Indian crust into the Himalayan wedge (Fig. 2B) (Bollinger et al. 2006),(Caldwell et al. 2013),(Gao et al. 2016). This is supported by the distribution of 40Ar/39Ar cooling ages (Wobus et al. 2006b). The third scenario takes into account that out-of-sequence thrusting occurs along the PT2 (Fig. 2C) (Wobus et al. 2005). In the course of the discussion we will interpret our results with respect to those theories and different structural settings to examine the tectonic convergence in the northwestern Himalaya.
Direction and velocity of plate motion Considering Eurasia as a fixed reference point the center of the Indian craton (GPS -station IISC in Bangalore) moves at a rate of 37±1mm/yr towards NNE (Wang et al. 2001). The analysis of GPS-measurements along the Himalaya range and in the Tibetan plateau revealed that the recent convergence is accommodated by multiple fault systems (England & Molnar 1997),(Wang et al. 2001). Furthermore, the convergence rates tend to decrease from the eastern to the western part of the mountain belt (Banerjee 2002; Stevens & Avouac 2015) and references in there). Geomorphological and structural studies have shown coincidently that shortening rates in the western parts of the mountain front are about 14±2mm∙yr- (Powers et al. 1998) which is about 20% less than in central Nepal (Lavé & Avouac 2000; Kundu et al. 2014).
The reconstruction of the Indian plate motion showed that India’s direction and velocity has changed during the last 20 Myr (Molnar & Stock 2009). Its rotating motion has changed the
13/62 direction from 10°N to 20°N and thus became clockwise (Molnar & Stock 2009). Moreover the convergence rate decreased from 44 to 34mm∙yr-1 in NW India (Molnar & Stock 2009) which is comparable with the latest results from Wang, 2001.
Seismic activity and geodetic velocities in NW India
In order to give a brief overview about the seismo-tectonic setting in the study area we use the seismic data from the NEIC catalogue (Fig. 3). This catalogue includes historic events since 1905 and a continuous record since 1973 for earthquakes with a moment magnitude larger than 3. The data reveal that the majority of the earthquakes in the southern part of the study area are located around the PT2. However, the amount of events decreases following this structural feature to the northwest. Just few events can be determined around the Kullu- Rampur Window. North of the tectonic window we see a large cluster of seismic events. Unfortunately, we have no information about the focal mechanisms but these events are assumed to belong to the extension in the Kaurik Chango Rift (Arora et al. 2012). Another noticeable accumulation of seismic events occurring on thrust faults can be seen around the Dhauladar Range and at the southern border of the Kishtwar window. Looking again further to the northwest just few events can be observed in the study area. The sparse amount of events is mainly located in the Kashmir Basin. In general, most of the events are located northeast of the MBT. Just very few events can be observed in area of the Subhimalaya.
The GPS velocities taken from a study of Banerjee, et al. draw a similar picture. The highest rates are measured by GPS stations in the hanging wall of the MBT (Banerjee et al. 2008). They fall in a range between 4-18 mm/yr while the lowest rates are measured by stations the southern study area. However, the velocities measured by stations in the Subhimalaya are seismically locked and therefore the slowest. They range between 0.5 and 4 mm/yr (Fig. 3).
14/62 Figure 3. Location and magnitude of seismic events from 1905 to 2016 with continuous record since 1973 (from NEIC catalogue). Moreover, red arrows illustrate the annual shortening rate measured by GPS stations in the northwestern Himalaya. (data from Banerjee et al.,2008).
These data show that we already have a good idea about how the recent convergence is accommodated in the Himalayan fault systems. However, these datasets cover a very short time span and do not include large seismic event with long recurrence times. Furthermore, we do not know which fault or fault segment accommodates the main shortening. Even if we determine the location of earthquake epicenters we cannot necessarily determine the ruptured fault. This is crucial for the understanding of the evolution and the propagation of the orogenic wedge. We use geomorphological methods in order to integrate the tectonic evolution over a larger period and to solve this lack of information.
15/62 Methods
River steepness and concavity index The profile of a river is the result of many environmental influences. Key factors responsible for its shape are tectonic and climatic conditions. Additionally, the profile shape strongly depends on the bedrock or underlying substrate, the most effective erosion process, the channel depth and width, the specific discharge and the periodicity of events. Because the complete erosional processes acing on a river basin are difficult to separate and quantify, many models use an empirical approach to describe fluvial erosion as a power law function of upstream area and channel slope. This approach is based on the assumption that erosion is primarily controlled by bed shear stress (Whipple & Tucker, 1999). The relationship between erosion, upstream area and slope can be approximated as follows:
E Sn
(Eq. 1) by Whipple and Tucker, 1999
This stream-power equation describes the erosion ( ) for a specific point in the river channel as a function of the erosion coefficient ( ), the upstream area ( ) which serves as a proxy for discharge ( ) (Whipple & Tucker, 1999) and the channel slope ( ) which is used as a proxy for erosional efficiency. The exponents and are scaling factors related to the hydraulic geometry, basin hydrology and erosion processes (Whipple & Tucker, 1999). Whipple and Tucker calculated that typical hydrological values result in a ⁄ ratio varying between 0.35 and 0.6 (Whipple & Tucker, 1999). By using Eq. 1 in combination with a known rock uplift rate , we can calculate the surface uplift rate (England & Molnar 1990). − − (Eq. 2) by Whipple and Tucker, 1999
In areas of topographic steady state, where the erosion rate equals the rock uplift rate, the surface elevation remains steady. Thus, Eq. 2 can be simplified and rearranged for the channel slope. For each point in the river profile, this equilibrium slope is described by the following equation:
16/62 -θ S A
(Eq. 3) by J.T. Hack, 1957 and J.J. Flint, 1977